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I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the stationary distribution of the process as $t\rightarrow\infty $ is Gamma-distributed. However, I'm interested in the distribution at a finite time $t$. Is there a known distribution (or transformation of a known distribution) that has this Laplace transform?

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